WASHINGTON—President Trump’s trade battle with China is shaping up as a definitive issue in the 2020 campaign, with shaken financial markets and fears of an economic slowdown testing the Republican’s insistence that his blunt-force trade posture will be an asset as he seeks re-election.

Democratic presidential hopefuls including front-runner Joe Biden already see vulnerabilities for Mr. Trump in the effects at home, including businesses and consumers hit by tariffs and farmers who can’t sell their crops to China because of...